Question:
If the ratio of the time periods of the electrons revolving in the first and \(n^{\text{th}}\) orbits of Hydrogen atom is \(1:64\), then the angular momentum of the electron in the \(n^{\text{th}}\) excited state of Hydrogen atom is \((h=\text{Planck's constant})\)
If the ratio of the time periods of the electrons revolving in the first and \(n^{\text{th}}\) orbits of Hydrogen atom is \(1:64\), then the angular momentum of the electron in the \(n^{\text{th}}\) excited state of Hydrogen atom is \((h=\text{Planck's constant})\)
Show Hint
For Hydrogen atom:
\[
T_n\propto n^3,
\]
\[
r_n\propto n^2,
\]
\[
v_n\propto \frac{1}{n},
\]
and
\[
L_n=\frac{nh}{2\pi}.
\]
Remember that the \(k^{\text{th}}\) excited state corresponds to orbit number \(k+1\).
Updated On: Jul 9, 2026
- \(\dfrac{3.5h}{\pi}\)
- \(\dfrac{5h}{\pi}\)
- \(\dfrac{2.5h}{\pi}\)
- \(\dfrac{2h}{\pi}\) \bigskip
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The Correct Option is C
Solution and Explanation
Concept:
In Bohr's model of Hydrogen atom,
\[
r_n \propto n^2,
\qquad
v_n \propto \frac{1}{n}.
\]
Therefore, the time period of revolution is
\[
T_n=\frac{2\pi r_n}{v_n}
\propto n^3.
\]
Also, the angular momentum in the \(n^{\text{th}}\) orbit is
\[
L_n=\frac{nh}{2\pi}.
\]
Step 1: Determine the orbit number \(n\). Given, \[ T_1:T_n=1:64. \] Since \[ T_n\propto n^3, \] \[ \frac{T_n}{T_1}=n^3=64. \] \[ n=4. \]
Step 2: Find the orbit corresponding to the \(n^{\text{th}}\) excited state. The \(n^{\text{th}}\) excited state corresponds to orbit number \[ n+1. \] Since \(n=4\), \[ \text{orbit number}=5. \]
Step 3: Calculate the angular momentum. \[ L=\frac{5h}{2\pi}. \] \[ L=\frac{2.5h}{\pi}. \]
Step 4: Write the final answer. \[ \boxed{L=\frac{5h}{2\pi}} \] \[ \boxed{L=\frac{2.5h}{\pi}} \] \[ \boxed{\text{Answer = (C)}} \]
Step 1: Determine the orbit number \(n\). Given, \[ T_1:T_n=1:64. \] Since \[ T_n\propto n^3, \] \[ \frac{T_n}{T_1}=n^3=64. \] \[ n=4. \]
Step 2: Find the orbit corresponding to the \(n^{\text{th}}\) excited state. The \(n^{\text{th}}\) excited state corresponds to orbit number \[ n+1. \] Since \(n=4\), \[ \text{orbit number}=5. \]
Step 3: Calculate the angular momentum. \[ L=\frac{5h}{2\pi}. \] \[ L=\frac{2.5h}{\pi}. \]
Step 4: Write the final answer. \[ \boxed{L=\frac{5h}{2\pi}} \] \[ \boxed{L=\frac{2.5h}{\pi}} \] \[ \boxed{\text{Answer = (C)}} \]
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